Effects of eddy currents in transformer windings
01 Aug 1966-Vol. 113, Iss: 8, pp 1387-1394
TL;DR: In this article, the effect of eddy currents on transformer windings is considered and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings.
Abstract: The effects of eddy currents in transformer windings are considered, and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings. The method consists in dividing the winding into portions, calculating the d.c. resistances and d.c. leakage inductances of each of these portions, and then multiplying the d.c. values by appropriate factors to obtain the corresponding a.c. values. These a.c. values are then referred to, say, the primary winding and summed to give the total winding resistance and leakage inductance of the transformer. Formulas are derived and quoted for calculating the d.c. resistances and leakage inductances of the winding portions. Theoretical expressions are derived for the variation with frequency etc. of the factors by which the d.c. values must be multiplied to obtain the corresponding a.c. values. These expressions are presented in the form of graphs, permitting the factors to be read as required.
Citations
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06 Nov 2009
TL;DR: In this paper, the area-product value of a resonant inductor is derived from the loaded-quality factor of a circuit, output power, and operating frequency assuming the window utilization factor and the maximum flax density.
Abstract: There are no well-established criteria for selecting the core for the design of resonant inductors. This paper presents new expressions of the area product for resonant inductors. By using proposed expressions, the area-product value can be calculated from loaded-quality factor of a resonant circuit, output power, and operating frequency assuming the window utilization factor and the maximum flax density. The area-product value expressed in terms of the loaded-quality factor is a good criterion for selecting the core. The design examples are given for single-wire winding and multiple-strand winding with a gapped core taking into account skin, proximity, and fringing effects.
70 citations
TL;DR: In this paper, a closed-form formula for calculating ohmic losses in switch-mode power supply (SMPS) transformers is presented, which is based on intensive two-dimensional (2D) finite element method (FEM) simulations.
Abstract: A new formula aimed at calculating ohmic losses in switch-mode power supply (SMPS) transformers is presented. It is based on intensive two-dimensional (2-D) finite element method (FEM) simulations, the results of which have been summarized in a closed-form formula following a "semi-empirical" approach. The main benefit of this new formula, specifically intended for industrial designers, is to combine the precision of 2-D models with the ease-of-use and speed of calculation of one-dimensional (1-D) models, on the whole frequency range. It accurately covers cases where the classical Dowell's formula significantly underestimated the losses, specifically those with significant edge effect in foil windings. Experimental validation and discussion of accuracy is provided. At the moment, the formula is only valid for one layer of foil located between a zero and a maximum of the magnetomotive force but a similar approach could be applied with success to other types of windings. Furthermore, the analytical expression proposed in the article, based on Maxwell equations, can be used as a stand-alone tool to model the real behavior of any type of winding. More accurate understanding of the 2-D field is also possible thanks to the direct link established between the losses and the geometrical data of the winding.
70 citations
TL;DR: In this paper, a two-stage dc-dc converter based on gallium nitride (GNT) was proposed, where the first regulated stage is a 2-phase interleaved buck converter and the second unregulated stage is an LLC (2-MHz) dc transformer.
Abstract: This paper develops a 200-W wide-input-range (64–160-to-24-V) rail grade dc–dc converter based on gallium nitride devices. A two-stage configuration is proposed. The first regulated stage is a two-phase interleaved buck converter ( $>$ 400 kHz), and the second unregulated stage is an LLC (2-MHz) dc transformer. In order to achieve high frequency and high efficiency, the critical-mode operation is applied for the buck converter, and the negative coupled inductors are used to reduce the frequency and the conduction losses. Then, a systematical methodology is proposed to optimize the planar-coupled inductors. For the unregulated LLC converter, it can always work at its most efficient point, and an analytical model is used to optimize the planar transformer. Finally, the proposed dc–dc converter, built in a quarter brick form factor, is demonstrated with a peak efficiency of 95.8% and a power density of 195 W/in $^3$ .
69 citations
TL;DR: In this paper, a semi-analytical approach for the analysis of the skin and proximity effects in multi-turn coils is proposed, where the complex permeability of a round conducting wire immersed in uniform time-harmonic magnetic fields is represented in a closed form.
Abstract: Native application of finite-element method (FEM) to the analysis of skin and proximity effects in multi-turn coils results in large equation systems, whose solution needs long computational time. This paper proposes a semi-analytical approach to overcome this problem. For the analysis of the proximity effect, the complex permeability of a round conducting wire immersed in uniform time-harmonic magnetic fields is represented in a closed form. Then, the homogenized complex permeability over the cross section of the multi-turn coil is analytically evaluated using the Ollendorff formula. The magnetoquasistatic problem is thus replaced by the magnetostatic one, in which the multi-turn coil is treated as a uniform material with the homogenized complex permeability. The skin effect is taken into consideration by introducing the corresponding impedance in the circuit equation. The proposed method is shown to give the impedance of multi-turn coils, which is in good agreement with that obtained by the conventional FEM as well as experiments.
69 citations
TL;DR: It is revealed that both the mission profile and the thermal cross-coupling effect have a significant impact on the prediction of system wear-out failure, and the dc-link electrolytic capacitor is the bottleneck of long-term reliability.
Abstract: In this paper, the wear-out performance of an impedance-source photovoltaic (PV) microinverter (MI) is evaluated and improved based on two different mission profiles. The operating principle and hardware implementation of the MI are first described. With the experimental measurements on a 300-W MI prototype and system-level finite-element method simulations, the electrothermal models are built for the most reliability-critical components, i.e., power semiconductor devices and capacitors. The dependence of the power loss on the junction/hotspot temperature is considered, the enclosure temperature is taken into account, and the thermal cross-coupling effect between components is modeled. Then, the long-term junction/hotspot temperature profiles are derived and further translated into components’ annual damages with the lifetime and damage accumulation models. After that, the Monte Carlo simulation and Weibull analysis are conducted to obtain the system wear-out failure probability over time. It reveals that both the mission profile and the thermal cross-coupling effect have a significant impact on the prediction of system wear-out failure, and the dc-link electrolytic capacitor is the bottleneck of long-term reliability. Finally, the multimode control with a variable dc-link voltage is proposed, and a more reliable dc-link electrolytic capacitor is employed, which results in a remarkable reliability improvement for the studied PV MI.
68 citations
References
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TL;DR: In this article, a multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers.
Abstract: IN any multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers. Eddy currents are set up in each layer in a direction to partly neutralize the magnetic intensities in the interior of the copper wire in each layer. As a result of the eddy-current losses in the copper, the effective resistance of the winding to the alternating current it carries may be many times its resistance to continuous currents.
103 citations
TL;DR: In this article, the authors discuss the more important causes of eddy currents in heavy conductors carrying alternating currents and surrounded on three sides by iron, and propose a method to identify the most important causes.
Abstract: The object of the present paper is the discussion of the more important causes of eddy currents in heavy conductors carrying alternating currents and surrounded on three sides by iron.
93 citations
64 citations
TL;DR: In this article, it is shown that a considerable proportion of the effective resistance of inductive coils when used at radio frequencies is caused by the eddy-currents set up in the wires of the coils by the alternating magnetic field in which they are situated, and that in extreme cases the alternating current resistance may amount to more than one hundred times the direct current resistance.
Abstract: It is well-known that a considerable proportion of the effective resistance of inductive coils when used at radio frequencies is caused by the eddy-currents set up in the wires of the coils by the alternating magnetic field in which they are situated, and that in extreme cases the alternating current resistance may amount to more than one hundred times the direct current resistance. It is therefore important to have reliable formulae for the eddy-current resistance of such coils in order to determine the conditions which will reduce the eddy-current losses to a minimum. The simplest case, that of a long straight cylindrical wire under the action of its own current, has been treated by Kelvin, Rayleigh, Heaviside, and others. The general effect is known as the “skin effect,” because the current tends to concentrate more and more upon the skin of the conductor as the frequency increases.
49 citations
TL;DR: In this article, the authors show how hyperbolic functions of complex angles may be applied to the solution of the problem of heat losses in rectangular conductors that are embedded in open slots.
Abstract: The principal object of this paper is to show how hyperbolic functions of complex angles may be applied to the solution of the problem of heat losses in rectangular conductors that are embedded in open slots. A certain knowledge of the functions themselves is presupposed. Inasmuch, however, as they are handled like trigometric functions of real angles?except in regard to the plus and minus signs?it is a simple matter to acquire the requisite technical skill to use them. The hyperbolic function of a complex angle, consisting as it does of a real and an imaginary part, may represent a vector?the real part being the component of the vector along the horizontal, and the imaginary part, component along the vertical. Thus, for example, A sinh (x + j x) represents a vector just as A e j ? A/?, A (cos ? + j sin ?) represent vectors. Considerable experience has shown that the vector method for handling a-c. problems is much superior to the original method in which simple trigonometric functions were used. With this lesson before us, it should require but little contact with the problem at hand to demonstrate the superiority of the vector method, even though it employs the possibly unfamiliar hyperbolic quantities. These hyperbolic vectors have been used for a number of years in the analysis of problems involving a-c. circuits, which have distributed inductance and capacitance, and have proved their usefulness.
27 citations